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Theorem brab1 4733
Description: Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
Assertion
Ref Expression
brab1 (𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
Distinct variable groups:   𝑧,𝐴   𝑧,𝑅
Allowed substitution hints:   𝐴(𝑥)   𝑅(𝑥)

Proof of Theorem brab1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3234 . . 3 𝑥 ∈ V
2 breq1 4688 . . . 4 (𝑧 = 𝑦 → (𝑧𝑅𝐴𝑦𝑅𝐴))
3 breq1 4688 . . . 4 (𝑦 = 𝑥 → (𝑦𝑅𝐴𝑥𝑅𝐴))
42, 3sbcie2g 3502 . . 3 (𝑥 ∈ V → ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥𝑅𝐴))
51, 4ax-mp 5 . 2 ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥𝑅𝐴)
6 df-sbc 3469 . 2 ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
75, 6bitr3i 266 1 (𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2030  {cab 2637  Vcvv 3231  [wsbc 3468   class class class wbr 4685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686
This theorem is referenced by: (None)
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